On the structure of continuous uninorms

Uninorms were introduced by Yager and Rybalov [13] as a generalization of triangular norms and conorms. We ask about properties of increasing, associative, continuous binary operation U in the unit interval with the neutral element e ∈ [0, 1]. If operation U is continuous, then e = 0 or e = 1. So, we consider operations which are continuous in the open unit square. As a result every associative, increasing binary operation with the neutral element e ∈ (0, 1), which is continuous in the open unit square may be given in [0, 1) or (0, 1] as an ordinal sum of a semigroup and a group. This group is isomorphic to the positive real numbers with multiplication. As a corollary we obtain the results of Hu, Li [7].